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   <h4 class="subsectionHead"><span class="titlemark">4.3   </span> <a 
 id="x28-920004.3"></a>2D SUPG/PSGP Elements</h4>
<!--l. 1909--><p class="noindent" >
   <h5 class="subsubsectionHead"><span class="titlemark">4.3.1   </span> <a 
 id="x28-930004.3.1"></a>Tr1SUPG element</h5>
<!--l. 1911--><p class="noindent" >Represents the linear triangular finite element for transient incompressible flow analysis using SUPG/PSPG
stabilization with equal order approximation of velocity and pressure fields. Each node has 3 degrees of freedoms (two
components of velocity and pressure). The node numbering is anti-clockwise. The element features are summarized in
Table&#x00A0;<a 
href="#x28-9300261">61<!--tex4ht:ref: Tr1SUPGsummary --></a>.
<!--l. 1917--><p class="indent" >   <hr class="figure"><div class="figure" 
>
                                                                                           
                                                                                           
<a 
 id="x28-9300126"></a>
                                                                                           
                                                                                           
<!--l. 1--><p class="noindent" ><img 
src="elementlibmanual-figure29.png" alt="PIC"  
>
<br /> <div class="caption" 
><span class="id">Figure&#x00A0;26: </span><span  
class="content">Tr1SUPG element. Node numbering, Side numbering and definition of local edge c.s.(a).</span></div><!--tex4ht:label?: x28-9300126 -->
                                                                                           
                                                                                           
<!--l. 1925--><p class="indent" >   </div><hr class="endfigure">
<div class="center" 
>
<!--l. 1928--><p class="noindent" >
<div class="table">
<!--l. 1928--><p class="noindent" ><a 
 id="x28-9300261"></a><hr class="float"><div class="float" 
> <!--tex4ht:inline--><div class="tabular"> <table id="TBL-63" class="tabular" 
cellspacing="0" cellpadding="0" rules="groups" 
><colgroup id="TBL-63-1g"><col 
id="TBL-63-1"></colgroup><colgroup id="TBL-63-2g"><col 
id="TBL-63-2"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-63-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-63-1-1"  
class="td11"><span 
class="cmbx-10">Keyword         </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-63-1-2"  
class="td11"><!--l. 1928--><p class="noindent" ><span 
class="cmbx-10">Tr1SUPG</span>                                                      </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-63-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-63-2-1"  
class="td11">Description          </td><td  style="white-space:wrap; text-align:left;" id="TBL-63-2-2"  
class="td11"><!--l. 1928--><p class="noindent" >linear triangular finite element for transient incompressible
flow analysis using SUPG/PSPG algorithm                    </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-63-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-63-3-1"  
class="td11">Specific parameters</td><td  style="white-space:wrap; text-align:left;" id="TBL-63-3-2"  
class="td11"><!--l. 1928--><p class="noindent" >[<span 
class="cmtt-10">vof</span>&#x00A0;#<span 
class="cmr-5">(rn)</span>] [<span 
class="cmtt-10">pvof</span>&#x00A0;#<span 
class="cmr-5">(rn)</span>]                                              </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-63-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-63-4-1"  
class="td11">Unknowns            </td><td  style="white-space:wrap; text-align:left;" id="TBL-63-4-2"  
class="td11"><!--l. 1929--><p class="noindent" >Two  velocity  components  (V_u  and  V_v)  and  pressure
(P_f) are required in each node.                                   </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-63-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-63-5-1"  
class="td11">Approximation      </td><td  style="white-space:wrap; text-align:left;" id="TBL-63-5-2"  
class="td11"><!--l. 1930--><p class="noindent" >Linear approximation of velocity and pressure fields.        </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-63-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-63-6-1"  
class="td11">Integration           </td><td  style="white-space:wrap; text-align:left;" id="TBL-63-6-2"  
class="td11"><!--l. 1931--><p class="noindent" >exact                                                                      </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-63-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-63-7-1"  
class="td11">Loads                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-63-7-2"  
class="td11"><!--l. 1933--><p class="noindent" >Constant  boundary  tractions  are  supported.  Body  loads
representing the self-weight load are supported.              </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-63-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-63-8-1"  
class="td11">Multi-fluid analysis</td><td  style="white-space:wrap; text-align:left;" id="TBL-63-8-2"  
class="td11"><!--l. 1944--><p class="noindent" >The  element  has  support  for  solving  problems  with  two
immiscible fluids in a fixed spatial domain. In the present
implementation, a VOF and LevelSet tracking algorithms
are  used  to  track  the  position  of  interface.  In  case  of
VOF  tracking,  an  initial  VOF  fraction  (volume  fraction
of reference fluid) can be specified using  <span 
class="cmtt-10">vof </span>(default is
zero). Element can also be marked as allways filled with
reference  fluid  (some  form  of  source)  using  parameter
<span 
class="cmtt-10">pvof </span>which  specifies  the  permanent  VOF value.  In  case
of LevelSet tracking, the initial levelset is specified using
reference  polygon  (see  corresponding  levelset  record  in
oofem input manual). The material model should be of type
<span 
class="cmbx-10">Keyword</span>:&#x00A0;<span 
class="cmtt-10">twofluidmat</span>, that supports modelling of two
immiscible fluids.                                                      </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-63-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-63-9-1"  
class="td11">Status                 </td><td  style="white-space:wrap; text-align:left;" id="TBL-63-9-2"  
class="td11"><!--l. 1945--><p class="noindent" >Reliable                                                                  </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-63-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-63-10-1"  
class="td11"> </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-63-11-"><td  style="white-space:nowrap; text-align:left;" id="TBL-63-11-1"  
class="td11">                </td></tr></table></div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;61: </span><span  
class="content">Tr1SUPG element summary</span></div><!--tex4ht:label?: x28-9300261 -->
</div><hr class="endfloat" />
</div>
</div>
   <h5 class="subsubsectionHead"><span class="titlemark">4.3.2   </span> <a 
 id="x28-940004.3.2"></a>Tr21SUPG element</h5>
<!--l. 1951--><p class="noindent" >Implementation of P2P1 Taylor Hood element for transient incompressible flow analysis using SUPG and LSIC
stabilization. It consists of globally continuous, piecewise quadratic functions for approximation in velocity space and
globally continuous, piecewise linear functions for approximation in pressure space. LBB condition is satisfied. There
are 3 degrees of freedom in vertices (two components of velocity and pressure), and 2 degrees of freedom in edge nodes
(two components of velocity only). The node numbering is anti-clockwise, vertices are numbered first. The element
features are summarized in Table&#x00A0;<a 
href="#x28-9400262">62<!--tex4ht:ref: Tr21SUPGsummary --></a>.
<!--l. 1959--><p class="indent" >   <hr class="figure"><div class="figure" 
>
                                                                                           
                                                                                           
<a 
 id="x28-9400127"></a>
                                                                                           
                                                                                           
<!--l. 2--><p class="noindent" ><img 
src="elementlibmanual-figure30.png" alt="PIC"  
>
<br /> <div class="caption" 
><span class="id">Figure&#x00A0;27: </span><span  
class="content">Tr21SUPG element - node and side numbering.</span></div><!--tex4ht:label?: x28-9400127 -->
                                                                                           
                                                                                           
<!--l. 1966--><p class="indent" >   </div><hr class="endfigure">
<div class="center" 
>
<!--l. 1968--><p class="noindent" >
<div class="table">
<!--l. 1968--><p class="noindent" ><a 
 id="x28-9400262"></a><hr class="float"><div class="float" 
> <!--tex4ht:inline--><div class="tabular"> <table id="TBL-64" class="tabular" 
cellspacing="0" cellpadding="0" rules="groups" 
><colgroup id="TBL-64-1g"><col 
id="TBL-64-1"></colgroup><colgroup id="TBL-64-2g"><col 
id="TBL-64-2"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-64-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-64-1-1"  
class="td11"><span 
class="cmbx-10">Keyword         </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-64-1-2"  
class="td11"><!--l. 1968--><p class="noindent" ><span 
class="cmbx-10">Tr21SUPG</span>                                                    </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-64-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-64-2-1"  
class="td11">Description          </td><td  style="white-space:wrap; text-align:left;" id="TBL-64-2-2"  
class="td11"><!--l. 1968--><p class="noindent" >P2P1 Taylor Hood element                                         </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-64-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-64-3-1"  
class="td11">Specific parameters</td><td  style="white-space:wrap; text-align:left;" id="TBL-64-3-2"  
class="td11"><!--l. 1968--><p class="noindent" >-                                                                            </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-64-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-64-4-1"  
class="td11">Unknowns            </td><td  style="white-space:wrap; text-align:left;" id="TBL-64-4-2"  
class="td11"><!--l. 1969--><p class="noindent" >Two  velocity  components  (V_u  and  V_v)  and  pressure
(P_f) in vertices and two velocity components (V_u and
V_v) in edge nodes are required.                                  </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-64-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-64-5-1"  
class="td11">Approximation      </td><td  style="white-space:wrap; text-align:left;" id="TBL-64-5-2"  
class="td11"><!--l. 1970--><p class="noindent" >Quadratic    approximation    of    velocity    and    linear
approximation of pressure fields.                                  </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-64-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-64-6-1"  
class="td11">Integration           </td><td  style="white-space:wrap; text-align:left;" id="TBL-64-6-2"  
class="td11"><!--l. 1973--><p class="noindent" >Integration  is  exact,  each  submatrix  of  element  stiffness
matrix  is  evaluated  in  proper  number  of  Gauss  points.
Submatrices connected with velocity are evaluated in 7 or
13 points, mixed velocity-pressure submatrices in 3 or 7
points, submatrices connected with pressure in 3 points.   </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-64-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-64-7-1"  
class="td11">Loads                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-64-7-2"  
class="td11"><!--l. 1975--><p class="noindent" >Constant  boundary  tractions  are  supported.  Body  loads
representing the self-weight load are supported.              </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-64-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-64-8-1"  
class="td11">Multi-fluid analysis</td><td  style="white-space:wrap; text-align:left;" id="TBL-64-8-2"  
class="td11"><!--l. 1977--><p class="noindent" >The element has no support for solving problems with two
immiscible fluids in a fixed spatial domain.                     </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-64-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-64-9-1"  
class="td11">Status                 </td><td  style="white-space:wrap; text-align:left;" id="TBL-64-9-2"  
class="td11"><!--l. 1978--><p class="noindent" >Reliable                                                                  </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-64-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-64-10-1"  
class="td11"> </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-64-11-"><td  style="white-space:nowrap; text-align:left;" id="TBL-64-11-1"  
class="td11">                </td></tr></table></div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;62: </span><span  
class="content">Tr21SUPG element summary</span></div><!--tex4ht:label?: x28-9400262 -->
</div><hr class="endfloat" />
</div>
</div>
   <h5 class="subsubsectionHead"><span class="titlemark">4.3.3   </span> <a 
 id="x28-950004.3.3"></a>Tr1SUPGAxi element</h5>
<!--l. 1984--><p class="noindent" >Represents the linear triangular finite element for transient incompressible flow analysis using SUPG/PSPG
stabilization with equal order approximation of velocity and pressure fields in 2d-axisymmetric setting.
Each node has 3 degrees of freedoms (two components of velocity and pressure). The y-axis is axis of
ratational symmetry. The node numbering is anti-clockwise. The element features are summarized in
Table&#x00A0;<a 
href="#x28-9500263">63<!--tex4ht:ref: Tr1SUPGAxisummary --></a>.
<!--l. 1990--><p class="indent" >   <hr class="figure"><div class="figure" 
>
                                                                                           
                                                                                           
<a 
 id="x28-9500128"></a>
                                                                                           
                                                                                           
<!--l. 1--><p class="noindent" ><img 
src="elementlibmanual-figure31.png" alt="PIC"  
>
<br /> <div class="caption" 
><span class="id">Figure&#x00A0;28: </span><span  
class="content">Tr1SUPGAxi element. Node numbering, Side numbering and definition of local edge c.s.(a).</span></div><!--tex4ht:label?: x28-9500128 -->
                                                                                           
                                                                                           
<!--l. 1998--><p class="indent" >   </div><hr class="endfigure">
<div class="center" 
>
<!--l. 2001--><p class="noindent" >
<div class="table">
<!--l. 2001--><p class="noindent" ><a 
 id="x28-9500263"></a><hr class="float"><div class="float" 
> <!--tex4ht:inline--><div class="tabular"> <table id="TBL-65" class="tabular" 
cellspacing="0" cellpadding="0" rules="groups" 
><colgroup id="TBL-65-1g"><col 
id="TBL-65-1"></colgroup><colgroup id="TBL-65-2g"><col 
id="TBL-65-2"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-65-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-65-1-1"  
class="td11"><span 
class="cmbx-10">Keyword         </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-65-1-2"  
class="td11"><!--l. 2001--><p class="noindent" ><span 
class="cmbx-10">Tr1SUPGAxi</span>                                                 </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-65-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-65-2-1"  
class="td11">Description          </td><td  style="white-space:wrap; text-align:left;" id="TBL-65-2-2"  
class="td11"><!--l. 2001--><p class="noindent" >linear equal order approximation axisymmetric element    </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-65-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-65-3-1"  
class="td11">Specific parameters</td><td  style="white-space:wrap; text-align:left;" id="TBL-65-3-2"  
class="td11"><!--l. 2001--><p class="noindent" >[<span 
class="cmtt-10">vof</span>&#x00A0;#<span 
class="cmr-5">(rn)</span>][<span 
class="cmtt-10">pvof</span>&#x00A0;#<span 
class="cmr-5">(rn)</span>]                                               </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-65-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-65-4-1"  
class="td11">Unknowns            </td><td  style="white-space:wrap; text-align:left;" id="TBL-65-4-2"  
class="td11"><!--l. 2002--><p class="noindent" >Two  velocity  components  (V_u  and  V_v)  and  pressure
(P_f) are required in each node.                                   </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-65-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-65-5-1"  
class="td11">Approximation      </td><td  style="white-space:wrap; text-align:left;" id="TBL-65-5-2"  
class="td11"><!--l. 2003--><p class="noindent" >Linear approximation of velocity and pressure fields.        </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-65-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-65-6-1"  
class="td11">Integration           </td><td  style="white-space:wrap; text-align:left;" id="TBL-65-6-2"  
class="td11"><!--l. 2004--><p class="noindent" >Gauss integration in seven point employed.                    </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-65-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-65-7-1"  
class="td11">Loads                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-65-7-2"  
class="td11"><!--l. 2006--><p class="noindent" >Constant  boundary  tractions  are  supported.  Body  loads
representing the self-weight load are supported.              </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-65-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-65-8-1"  
class="td11">Multi-fluid analysis</td><td  style="white-space:wrap; text-align:left;" id="TBL-65-8-2"  
class="td11"><!--l. 2016--><p class="noindent" >The  element  has  support  for  solving  problems  with  two
immiscible fluids in a fixed spatial domain. In the present
implementation,  a  VOF  tracking  algorithm  is  used  to
track  the  position  of  interface.  An  initial  VOF  fraction
(volume fraction of reference fluid) can be specified using
<span 
class="cmtt-10">vof </span>(default  is  zero).  Element  can  also  be  marked  as
always  filled  with  reference  fluid  (some  form  of  source)
using parameter <span 
class="cmtt-10">pvof </span>which specifies the permanent VOF
value. In this case, the material model should be of type
<span 
class="cmbx-10">Keyword</span>:&#x00A0;<span 
class="cmtt-10">twofluidmat</span>, that supports modelling of two
immiscible fluids.                                                      </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-65-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-65-9-1"  
class="td11">Status                 </td><td  style="white-space:wrap; text-align:left;" id="TBL-65-9-2"  
class="td11"><!--l. 2018--><p class="noindent" >                                                 </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-65-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-65-10-1"  
class="td11"> </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-65-11-"><td  style="white-space:nowrap; text-align:left;" id="TBL-65-11-1"  
class="td11">                </td></tr></table></div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;63: </span><span  
class="content">Tr1SUPGAxi element summary</span></div><!--tex4ht:label?: x28-9500263 -->
</div><hr class="endfloat" />
</div>
</div>
                                                                                           
                                                                                           
                                                                                           
                                                                                           
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